60 research outputs found
MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings
A finite ring R and a weight w on R satisfy the Extension Property if every
R-linear w-isometry between two R-linear codes in R^n extends to a monomial
transformation of R^n that preserves w. MacWilliams proved that finite fields
with the Hamming weight satisfy the Extension Property. It is known that finite
Frobenius rings with either the Hamming weight or the homogeneous weight
satisfy the Extension Property. Conversely, if a finite ring with the Hamming
or homogeneous weight satisfies the Extension Property, then the ring is
Frobenius.
This paper addresses the question of a characterization of all bi-invariant
weights on a finite ring that satisfy the Extension Property. Having solved
this question in previous papers for all direct products of finite chain rings
and for matrix rings, we have now arrived at a characterization of these
weights for finite principal ideal rings, which form a large subclass of the
finite Frobenius rings. We do not assume commutativity of the rings in
question.Comment: 12 page
The extension problem for Lee and Euclidean weights
The extension problem is solved for the Lee and Euclidean weights over three families of rings of the form : , , or with and prime. The extension problem is solved for the Euclidean PSK weight over for all
Equivalence Theorems and the Local-Global Property
In this thesis we revisit some classical results about the MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight-preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study a local-global property that naturally generalizes the MacWilliams equivalence theorems. Making use of F-partitions, we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid
Self-Dual Codes over Commutative Frobenius Rings
We prove that self-dual codes exist over all finite commutative Frobenius rings, via their decomposition by the Chinese Remainder Theorem into local rings. We construct non-free self-dual codes under some conditions, using self-dual codes over finite fields, and we also construct free self-dual codes by lifting elements from the base finite field. We generalize the building-up construction for finite commutative Frobenius rings, showing that all self-dual codes with minimum weight greater than 2 can be obtained in this manner in cases where the construction applies. Key Words: Self-dual codes, codes over rings. 2
Some combinatorial invariants determined by Betti numbers of Stanley-Reisner ideals
The papers in this thesis are not available in Munin:
Paper 1: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'Betti numbers associated to
the facet ideal of a matroid', available in Bulletin of the Brazilian Mathematical Society 45 no. 4, 727-744
Paper 2: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'A generalization of weight
polynomials to matroids', manuscript
Paper 3: Jan Roksvold, Hugues Verdure (2015): 'Betti numbers of skeletons', manuscriptThe thesis contains new results on the connection between the algebraic properties of certain ideals of a polynomial ring and properties of error-correcting linear codes, matroids and simplicial complexes. We demonstrate that the graded Betti numbers of the facet ideal of a matroid are determined by the Betti numbers of the blocks of the matroid. The extended weight enumerator of coding theory is generalized to matroids. We show that this generalization is equivalent to the Tutte-polynomial, and that the coefficients of this polynomial is determined by Betti numbers of the Stanley-Reisner ideal of the matroid and its elongations. The Betti numbers of the Stanley-Reisner ring of a skeleton of a simplicial complex is demonstrated to be an integral linear combination of the Betti numbers associated to the original complex
A transform approach to polycyclic and serial codes over rings
Producción CientÃficaIn this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.Ministerio de Ciencia, Innovación y Universidades / Agencia Estatal de Investigación / 0.13039/501100011033 (grant PGC2018-096446-B-C21
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